Local Perturbations Of A Fluke Switch Point For Intensive Rent

Figure 1: A Parameter Space

1.0 Introduction

This is a re-creation and elaboration of a previous post.

The analysis of the choice of technique, in models of circulating and fixed capital, can be based on the construction of a wage-rate of profits frontier. Given a technology in which requirements for use can be satisfied, prices of production for a feasible technique, including the wage, are uniquely determined by the given rate of profits. If the rate of profits is in a range where such prices are non-negative for at least one technique, one of the techniques is uniquely cost-minimizing, except at switch points. These properties do not necessarily hold in models of general joint production. An examination of local perturbations in an example of intensive rent illustrates surprising possibilities.

2.0 Technology

Table 1 presents coefficients of production, a perturbation of an example from D'Agata (1983). Only one type of land exists, and three processes are known for producing corn on it. The scarcity of land is shown by the possibility of two corn-producing processes being operated side-by-side in the cost-minimizing technique.

Table 1: Coefficients of Production

Inputs	Industry
	Iron	Steel	Corn
	I	II	III	IV	V
Labor	1	1	1	57/20	21/20
Land	0	0	1	13/10	21/20
Iron	0	0	1/10	a1,4	a1,5
Steel	0	0	2/5	13/100	21/200
Corn	1/10	3/5	1/10	2/5	21/50

Following D'Agata, assume that one hundred acres of land are available and that net output consists of 90 tons iron, 60 tons steel, and 19 bushels corn. The net output is also the numeraire. All three commodities must be produced for any composition of net output. Table 2 lists the available techniques. Only Alpha, Delta, and Epsilon are feasible for the parameter ranges considered. Not all land is farmed and only one corn-producing process is operated under Alpha. Two corn-producing processes are operated together under Delta and Epsilon.

Table 2: Techniques of Production

Technique	Processes
Alpha	I, II, III
Beta	I, II, IV
Gamma	I, II, V
Delta	I, II, III, IV
Epsilon	I, II, III, V
Zeta	I, II, IV, V

3.0 A Fluke Switch Point

Figure 2 shows the wage and rent curves for feasible techniques at a selected parametrization. I take the wage curve for a technique to be defined only for non-negative rates of profits at which the wage, rent per acre, and the prices of produced commodities are non-negative. The wage is negative for Delta for rates of profits below that at the switch point, and rent is negative for rates of profits greater than that at the switch point. Rent is negative for Epsilon for rates of profits less than at the switch point, and the wage is negative for greater rates of profits. Thus, the switch point is the only point on the wage curves for Delta and Epsilon. The switch point is a fluke in at least two ways. It is a switch point for three techniques, not two. And it is on the axis for the rate of profits.

Figure 2: Wage and Rent Curves for a Fluke Case

4.0 Local Perturbations within a Parameter Sapce

Figure 1, at the top of this post, shows a partition of the parameter space around this fluke case. An intersection of three wage curves over the axis for the rate of profits is a combination of three pairs of wage curves intersecting over the axis for the rate of profits. These three fluke cases are the partitions between regions 1 and 2, regions 2 and 3, regions 6 and 7, regions 7 and 8, and regions 8 and 1. The partition between regions 3 and 4 is associated with the fluke case of three wage curves intersecting at a non-negative rate of profits. The partitions between regions 4 and 5 and between regions 5 and 6 illustrate a fluke switch point specific to models of rent.

Regions 2 through 8 illustrate the possible non-uniqueness and non-existence of a cost-minimizing technique. For concreteness, consider the point in region 5 with the wage curves and variation in rent per acre illustrated in Figure 3. For rates of profits up to the first switch point, Alpha is cost-minimizing. Epsilon is cost-minimizing between the switch points, and Delta is also cost-minimizing for high rates of profits in this range. Beyond the second switch point, no technique is cost-minimizing. Whether or not land is scarce depends on the distribution of income.

4.1 The Choice of Technique in Region 5

Figure 3: Wage and Rent Curves in Region 5

How can one determine which techniques are cost-minimizing for a given rate of profits? Given the technique and the rate of profits, the costs of the capital goods, the rent on land, and wages can be summed for a unit level for each process. Iron, steel, and corn inputs incur the going rate of profits in this sum. The difference between the revenues and this sum is the extra profits obtained in operating a process. By definition, no process comprising the technique yields extra profits. The technique is cost-minimizing if extra profits cannot be obtained by operating any other process

For the parameters illustrated in Figure 3, extra profits are obtained by operating process IV or V at Alpha prices for a rate of profits greater than that at the first switch point. Alpha is only cost minimizing at a lower rate of profits. Figure 4 depicts the extra profits available from the last two corn-producing process at Delta and Epsilon prices. The range of rates of profits in which each technique is cost-minimizing is indicated, and these ranges overlap. For rates of profits immediately greater than the rate of profits at the second switch point, prices of production indicate that Epsilon should be adopted when prices of production for Delta prevail, and that Delta should be adopted when prices of production for Epsilon prevail. This circuit is a manifestation of the non-existence of a cost-minimizing technique.

Figure 4: Extra Profits for Delta and Epsilon in Region 5

4.2 Fluke Cases Bordering Region 5

Presenting two fluke switch points might assist in understanding how the analysis of the choice of technique varies in the part of the parameter space examined here. Figure 5 shows the wage and rent curves for a fluke switch point for parameters on the partition between regions 4 and 5. This fluke switch point is associated with the disapperance of the range of the rate of profits, in region 4, where both Alpha and Epsilon are cost-minimizing. It is associated with the emergence, in region 5, of a range of the rate of profits where only Epsilon is cost-minimizing.

Figure 5: A Fluke Switch Point on the Upper Boundary of Region 5

Figure 6, on the other hand, shows a fluke switch point for parameters on the boundary of regions 5 and 6. This switch point is associated with the disappearance of a range of the rate of profits where only Alpha and Epsilon have defined wage and rent curves and neither technique is cost-minimizing. And it is associated with the appearance of a range of the rate of profits where only Alpha and Delta have defined wage and rent curves and neither technique is cost-minimizing. The fluke switch points in Figure 5 and Figure 6 can only arise in models of joint production, including models of rent.

Figure 6: A Fluke Switch Point on the Lower Boundary of Region 5

4.3 Overview of Regions as a Whole

Qualitative properties of the analysis of the choice of technique do not vary within each numbered region. Table 3 describes the variation in the cost-minimizing technique with the rate of profits in each numbered region in Figure 1. In region 1, Alpha is cost-minimizing for all feasible rates of profits. Land is not scarce, and obtains no rent. For a high enough rate of profits in region 2, Alpha and Delta are both non-uniquely cost-minimizing. The wage curve for Delta slopes up and rent per acre decreases with the rate of profits when Delta is operated. The switch point for Alpha and Delta is at a positive wage. For any rate of profits greater than the rate of profits at the switch point, no cost-minimizing technique exists. In region 3, a switch point between Alpha and Epsilon occurs at a rate of profits higher than the maximum rate of profits for Delta. Epsilon is never cost-minimizing.

Table 3: Ranges of the Rate of Profits by Region

Region	Range	Technique	Comment
1	0 ≤ r ≤ rα,max	Alpha	Delta and Epsilon have negative wage or rent throughout.
2	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
3	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rε,min	None	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
4	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rε,min	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rε,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
5	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
6	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rε,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rε,max ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
7	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rδ,min	None	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
8	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.

In region 4, a switch point exists on the wage frontier between Alpha and Epsilon, at a rate of profits greater than the minimum rate of profits for Delta. A range of the rate of profits remains at which Alpha and Delta are both non-uniquely cost-minimizing. Above the rate of profits at this switch point, the wage frontier resembles the wage frontier in Figure 3 at rates of profits greater than the minimum rate of profits for Delta. In region 5, the range of the rate of profits at which both Alpha and Delta are cost-minimizing has disappeared. In region 6, the range of the rate of profits has disappeared in which no technique is cost minimizing, but Epsilon has a positive rate of profits and rent and Delta does not.

In region 7, Delta is no longer cost-minimizing at any feasible rate of profits. Alpha is cost-minimizing at a low rate of profits, and Epsilon is uniquely cost-minimizing at any feasible rate of profits greater than the rate of profits at the switch point between Alpha and Epsilon. In region 8, Delta is not only no longer ever cost-minimizing, but Delta never has both a positive rate of profits and rent.

Suppose one is not interested in qualitative variations in ranges of the rate of profits in which no cost-minimizing technique exists. In one range for some regions, Alpha has a positive rate of profits, and Delta and Epsilon each have positive rates of profits and a positive rent. Yet when prices for Delta prevail, extra profits can be obtained by operating processes in Epsilon. And when prices for Epsilon prevail, extra profits come from adopting Delta. In another range of rates of profits, neither Delta nor Epsilon obtain both non-negative rates of profits and rents. Yet Alpha is not cost-minimizing. Ignoring these variations, regions 2 and 3 can be combined. Regions 5 and 6 can be combined. Likewise, regions 7 and 8 can be combined

5.0 Conclusion

Whether or not land obtains a rent can depend on the distribution of income. For a low-enough rate of profits in regions 2 through 8, the first three processes are operated. Iron, steel, and corn are each produced with one process, and land obtains no rent. For a higher rate of profits, the Delta or Epsilon technique can be cost-minimizing. Corn is produced by two processes, and scarce land obtains a rent. Even if the requirements for use can feasibly be satisfied with some land not farmed, the cost-minimizing technique may be such that two processes are operated side-by-side on land, with no land lying fallow. The example illustrates that an examination of fluke switch points can help in understanding qualitative variations in the analysis of the choice of technique, even in a case where certain properties of models of circulating capital do not hold.

References

• D'Agata, Antonio. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica, 35: 147-158.
• Kurz, Heinz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.